K11a82

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K11a81

K11a83

1 Knot presentations
2 Three dimensional invariants
3 Four dimensional invariants
4 Polynomial invariants
5 "Similar" Knots (within the Atlas)
6 Vassiliev invariants
7 Khovanov Homology
8 Computer Talk
9 Modifying This Page

(Knotscape image)

See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a82's page at Knotilus!

Visit K11a82's page at the original Knot Atlas!

[edit K11a82 Quick Notes]

[edit K11a82 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,4,11,3 X_12,5,13,6 X_16,8,17,7 X_2,10,3,9 X_22,11,1,12 X_18,13,19,14 X_20,15,21,16 X_6,18,7,17 X_14,19,15,20 X_8,21,9,22
Gauss code	1, -5, 2, -1, 3, -9, 4, -11, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, -8, 11, -6
Dowker-Thistlethwaite code	4 10 12 16 2 22 18 20 6 14 8

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a82/ThurstonBennequinNumber
Hyperbolic Volume	13.8883
A-Polynomial	See Data:K11a82/A-polynomial

[edit Notes for K11a82's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	2

[edit Notes for K11a82's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 5 t 3 -12 t 2 + 19 t -21 + 19 t -1 -12 t -2 + 5 t -3 - t -4$
Conway polynomial	$- z 8 -3 z 6 -2 z 4 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 95, -2 }
Jones polynomial	$- q 4 + 3 q 3 -6 q 2 + 10 q -12 + 15 q -1 -15 q -2 + 13 q -3 -10 q -4 + 6 q -5 -3 q -6 + q -7$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + a 4 z 6 -6 a 2 z 6 + 2 z 6 + 4 a 4 z 4 -14 a 2 z 4 - z 4 a -2 + 9 z 4 + 5 a 4 z 2 -15 a 2 z 2 -3 z 2 a -2 + 13 z 2 + 2 a 4 -6 a 2 -2 a -2 + 7$
Kauffman polynomial (db, data sources)	$a 2 z 10 + z 10 + 4 a 3 z 9 + 7 a z 9 + 3 z 9 a -1 + 7 a 4 z 8 + 11 a 2 z 8 + 3 z 8 a -2 + 7 z 8 + 7 a 5 z 7 -15 a z 7 -7 z 7 a -1 + z 7 a -3 + 5 a 6 z 6 -14 a 4 z 6 -43 a 2 z 6 -12 z 6 a -2 -36 z 6 + 3 a 7 z 5 -12 a 5 z 5 -21 a 3 z 5 -6 a z 5 -4 z 5 a -1 -4 z 5 a -3 + a 8 z 4 -5 a 6 z 4 + 14 a 4 z 4 + 51 a 2 z 4 + 15 z 4 a -2 + 46 z 4 -3 a 7 z 3 + 11 a 5 z 3 + 28 a 3 z 3 + 22 a z 3 + 13 z 3 a -1 + 5 z 3 a -3 - a 8 z 2 + a 6 z 2 -7 a 4 z 2 -27 a 2 z 2 -8 z 2 a -2 -26 z 2 -4 a 5 z -10 a 3 z -10 a z -6 z a -1 -2 z a -3 + 2 a 4 + 6 a 2 + 2 a -2 + 7$
The A2 invariant	$q 20 - q 18 + 2 q 16 - q 14 - q 12 + q 10 -4 q 8 + 2 q 6 -2 q 4 + 2 q 2 + 3 + 3 q -4 - q -6 - q -12$
The G2 invariant	$q 114 -2 q 112 + 4 q 110 -6 q 108 + 5 q 106 -3 q 104 -2 q 102 + 10 q 100 -17 q 98 + 24 q 96 -27 q 94 + 19 q 92 -7 q 90 -13 q 88 + 37 q 86 -55 q 84 + 66 q 82 -62 q 80 + 39 q 78 - q 76 -44 q 74 + 91 q 72 -119 q 70 + 122 q 68 -90 q 66 + 26 q 64 + 56 q 62 -124 q 60 + 166 q 58 -152 q 56 + 86 q 54 + 9 q 52 -103 q 50 + 148 q 48 -128 q 46 + 50 q 44 + 54 q 42 -134 q 40 + 144 q 38 -84 q 36 -38 q 34 + 161 q 32 -239 q 30 + 217 q 28 -113 q 26 -48 q 24 + 202 q 22 -291 q 20 + 282 q 18 -182 q 16 + 24 q 14 + 130 q 12 -230 q 10 + 246 q 8 -165 q 6 + 40 q 4 + 89 q 2 -159 + 157 q -2 -70 q -4 -45 q -6 + 148 q -8 -183 q -10 + 139 q -12 -24 q -14 -108 q -16 + 209 q -18 -228 q -20 + 170 q -22 -58 q -24 -69 q -26 + 156 q -28 -184 q -30 + 152 q -32 -79 q -34 - q -36 + 56 q -38 -81 q -40 + 72 q -42 -47 q -44 + 19 q -46 + 3 q -48 -15 q -50 + 14 q -52 -11 q -54 + 6 q -56 -2 q -58 + q -60$

Further Quantum Invariants

K11a82 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["K11a82"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $- t 4 + 5 t 3 -12 t 2 + 19 t -21 + 19 t -1 -12 t -2 + 5 t -3 - t -4$

In[5]:= Conway[K][z]

Out[5]= $- z 8 -3 z 6 -2 z 4 + 1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= ${1}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 95, -2 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $- q 4 + 3 q 3 -6 q 2 + 10 q -12 + 15 q -1 -15 q -2 + 13 q -3 -10 q -4 + 6 q -5 -3 q -6 + q -7$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $- a 2 z 8 + a 4 z 6 -6 a 2 z 6 + 2 z 6 + 4 a 4 z 4 -14 a 2 z 4 - z 4 a -2 + 9 z 4 + 5 a 4 z 2 -15 a 2 z 2 -3 z 2 a -2 + 13 z 2 + 2 a 4 -6 a 2 -2 a -2 + 7$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $a 2 z 10 + z 10 + 4 a 3 z 9 + 7 a z 9 + 3 z 9 a -1 + 7 a 4 z 8 + 11 a 2 z 8 + 3 z 8 a -2 + 7 z 8 + 7 a 5 z 7 -15 a z 7 -7 z 7 a -1 + z 7 a -3 + 5 a 6 z 6 -14 a 4 z 6 -43 a 2 z 6 -12 z 6 a -2 -36 z 6 + 3 a 7 z 5 -12 a 5 z 5 -21 a 3 z 5 -6 a z 5 -4 z 5 a -1 -4 z 5 a -3 + a 8 z 4 -5 a 6 z 4 + 14 a 4 z 4 + 51 a 2 z 4 + 15 z 4 a -2 + 46 z 4 -3 a 7 z 3 + 11 a 5 z 3 + 28 a 3 z 3 + 22 a z 3 + 13 z 3 a -1 + 5 z 3 a -3 - a 8 z 2 + a 6 z 2 -7 a 4 z 2 -27 a 2 z 2 -8 z 2 a -2 -26 z 2 -4 a 5 z -10 a 3 z -10 a z -6 z a -1 -2 z a -3 + 2 a 4 + 6 a 2 + 2 a -2 + 7$

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a7, K11a33,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a33,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["K11a82"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { $- t 4 + 5 t 3 -12 t 2 + 19 t -21 + 19 t -1 -12 t -2 + 5 t -3 - t -4$ , $- q 4 + 3 q 3 -6 q 2 + 10 q -12 + 15 q -1 -15 q -2 + 13 q -3 -10 q -4 + 6 q -5 -3 q -6 + q -7$ }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {10_116, K11a7, K11a33,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {K11a33,}

[edit] Vassiliev invariants

V₂ and V₃:

(0, 2)

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-2 is the signature of K11a82. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-3

-2

-1

-3

-5

-2

-7

-9

-4

-11

-13

-2

-15

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$